SymTFT construction of gapless exotic-foliated dual models

We construct Symmetry Topological Field Theories (SymTFTs) for continuous subsystem symmetries, which are inherently non-Lorentz-invariant. Our framework produces dual bulk descriptions—gapped foliated and exotic SymTFTs—that generate gapless boundary theories with spontaneous subsystem symmetry breaking via interval compactification. In analogy with the sandwich construction of SymTFT, we call this Mille-feuille. This is done by specifying gapped and symmetry-breaking boundary conditions. In this way we obtain the foliated dual realizations of a topological theory enjoying subsystem symmetries as well, i.e. a foliated BF-type theory for various models, including the XY plaquette, XYZ cube, and $\phi$, $\hat{\phi}$ theories. This provides a systematic method for generating free theories that non-linearly realize subsystem symmetries.